Chabauty Without the Mordell-Weil Group

  • Michael Stoll


Based on ideas from recent joint work with Bjorn Poonen, we describe an algorithm that can in certain cases determine the set of rational points on a curve C, given only the p-Selmer group S of its Jacobian (or some other abelian variety C maps to) and the image of the p-Selmer set of C in S. The method is more likely to succeed when the genus is large, which is when it is usually rather difficult to obtain generators of a finite-index subgroup of the Mordell-Weil group, which one would need to apply Chabauty’s method in the usual way. We give some applications, for example to generalized Fermat equations of the form x5 + y5 = z p .


Rational points on curves Chabauty’s method Selmer group 

Subject Classification

11G30 14G05 14G25 14H25 11Y50 11D41 



I would like to thank Bjorn Poonen for useful discussions and MIT for its hospitality during a visit of two weeks in May 2015, when these discussions took place. All computations were done using the computer algebra system Magma [3].


  1. 1.
    J.S. Balakrishnan, R.W. Bradshaw, K.S. Kedlaya, Explicit Coleman integration for hyperelliptic curves, in Algorithmic Number Theory. Lecture Notes in Computer Science, vol. 6197 (Springer, Berlin, 2010), pp. 16–31Google Scholar
  2. 2.
    M. Bhargava, B.H. Gross, The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, in Automorphic Representations and L-Functions. Tata Institute of Fundamental Research Studies in Mathematics, vol. 22 (Tata Institute of Fundamental Research, Mumbai, 2013), pp. 23–91Google Scholar
  3. 3.
    W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    N. Bruin, E.V. Flynn, Exhibiting SHA[2] on hyperelliptic Jacobians. J. Number Theory 118, 266–291 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    N. Bruin, M. Stoll, Two-cover descent on hyperelliptic curves. Math. Comput. 78, 2347–2370 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    N. Bruin, M. Stoll, The Mordell-Weil sieve: proving non-existence of rational points on curves. LMS J. Comput. Math. 13, 272–306 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    S.R. Dahmen, S. Siksek, Perfect powers expressible as sums of two fifth or seventh powers. Acta Arith. 164, 65–100 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    N. Freitas, B. Naskrȩcki, M. Stoll, The generalized Fermat equation with exponents 2, 3, n. Preprint (2017). arXiv:1703.05058 [math.NT]Google Scholar
  9. 9.
    W. Ho, A. Shankar, I. Varma, Odd degree number fields with odd class number. Preprint (2016). arXiv:1603.06269Google Scholar
  10. 10.
    W.G. McCallum, On the method of Coleman and Chabauty. Math. Ann. 299, 565–596 (1994)MathSciNetCrossRefGoogle Scholar
  11. 11.
    B. Poonen, M. Stoll, Most odd degree hyperelliptic curves have only one rational point. Ann. Math. (2) 180, 1137–1166 (2014)Google Scholar
  12. 12.
    E.F. Schaefer, 2-Descent on the Jacobians of hyperelliptic curves. J. Number Theory 51, 219–232 (1995)MathSciNetCrossRefGoogle Scholar
  13. 13.
    E.F. Schaefer, M. Stoll, How to do a p-descent on an elliptic curve. Trans. Am. Math. Soc. 356, 1209–1231 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith. 98, 245–277 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M. Stoll, Independence of rational points on twists of a given curve. Compos. Math. 142, 1201–1214 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    M. Stoll, Finite descent obstructions and rational points on curves. Algebra Number Theory 1, 349–391 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Stoll, Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank. J. Eur. Math. Soc. Preprint. arXiv:1307.1773 (to appear)Google Scholar
  18. 18.
    R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. Math. (2) 141, 553–572 (1995)Google Scholar
  19. 19.
    The On-Line Encyclopedia of Integer Sequences,
  20. 20.
    A. Venkatesh, J.S. Ellenberg, Statistics of number fields and function fields, in Proceedings of the International Congress of Mathematicians, vol. II (Hindustan Book Agency, New Delhi, 2010)Google Scholar
  21. 21.
    A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. Math. (2) 141, 443–551 (1995)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany

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